Engaging students: Finding the inverse, converse, and contrapositive

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Heidee Nicoll. Her topic, from Geometry: finding the inverse, converse, and contrapositive.

How could you as a teacher create an activity or project that involves your topic?

I would start this lesson with if-then statements that were not math related. I would use simple examples such as “If it is raining, then my mother will not let me play outside.” Students will be in groups, and will each group will have a set of cutouts, with each set containing two copies of the word “not”, a card with “if” and a card with “then,” and each “if” statement and each “then” statement on separate cards. They will also have a worksheet that gives them space to write the sentences that we come up with as a class. As the teacher, I will have a set of cutouts that will have either magnets or tape on the back that I will have on the board. I will show them an example, before having them work on their own. I will have the cards, for example “If” “It is raining” “then” “my mother will not let me play outside” on the board. Then I will put a “not” card in front of each statement and ask the students what this statement means. It will say “If” “not” “it is raining” “then” “not” “my mother will not let me play outside,” which translates to “If it is not raining, my mother will let me play outside.” The students will copy the grammatically correct statement onto their worksheet. I will ask them if it is a true statement. Then, I will put the statement back in its original form, and then will switch the “if” and “then” statements, which would result in “If” “my mother will not let me play outside” “then” “it is raining.” The students will copy down this sentence and will discuss whether or not it is true. Lastly, we will do the contrapositive of the statement, and switch the “if” and “then” statements and add the “not” cards. The students will then do several sentences on their own, moving around the cards to form the statements, copying the sentences onto their worksheets, and talking as a group about whether or not the statements are true. This will help the students see the concept behind these different statements before having to learn the names inverse, converse, and contrapositive, and without having to think about them in terms of geometry.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

This clip from some Star Trek episode shows an example of times in the English language when it might be hard to decode exactly what someone is saying because of the word “not” or the use of double negatives.

I would show the students the clip and ask them what the man meant by “nobody helps nobody but himself” and if that was a true statement. If they decide that it is not true, then I would ask them what they would change about the sentence to make it true. Although this clip does not explicitly use the ideas of inverse, converse, or contrapositive, it shows the importance of being able to take a somewhat confusing or ambiguous statement and understand it logically. In order for students to understand inverse, converse, and contrapositive, they need to understand that the phrase “this is not an odd number” also means “this is an even number,” or that “this polygon does not have an uneven number of sides” means that “this polygon has an even number of sides.” I would show them examples such as these, and have the students share what they think the statements mean. We would have a class discussion about how language can be confusing at times, and how we need to be able to decode it.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

I would use Kahoot! to create an online quiz. I would have questions such as “which of these statements is logically true?” and “which of these statements is logically false?” Each answer choice would be a short statement, some math related, such as “if a number ends in 2 it is even” and some not related, such as “if the sun is out today, then it is warm outside.” I would also include statements that were the inverse, converse, or contrapositive, such as “if it is not warm outside, then the sun is not out today.” The students would have to read all the answer choices and pick the one that was true or false, depending on what the question was asking. This would get them thinking about whether or not certain statements are true, and would give them practice logically decoding words and phrases. Kahoot! keeps track of the students that answer correctly and quickly and keeps points, so it would be a small competition, which students normally enjoy.